How to Reason about Self-Locating Belief

Transcription

1 How to Reason about Self-Locating Belief David Shulman Department of Mathematics Delaware Valley College July 5, 2011 Abstract When reasoning about self-locating belief, one should reason as if one were a randomly selected bit of information. This principle can be considered to be an application of Bostrom s Strong Self-Sampling Assumption(SSSA)[2] according to which one should reason as if one were a randomly selected element of some suitable reference class of observer-moments. The reference class is the class of all observermoments. In order to randomly select an observer-moment from the reference class, one first randomly chooses a possible world w and then selects an observer-moment z from world w. The probability that one selects z given that one has chosen w should be proportional to the amount of information that z is capable of representing. There are both wagering arguments and relative frequency arguments that support our theory of anthropic reasoning. Our theory works best when the amount of information represented is finite. The infinite case is represented as a limit of a finite cases. We can learn from experience how best to represent the infinite case as a limit of finite cases and also learn from experience whether our theory or some other theory is the superior theory of anthropic reasoning. In order to test which theory is best, we use standard Bayesian methodology: We just need prior probabilities for the theories that are being tested and then we only have to use Bayes rule. 1 Reasoning about Self-Locating Belief In this paper, we describe a general, Bayesian theory of anthropic reasoning. We use the expressions anthropic reasoning and reasoning about self-locating belief to refer to reasoning about our identity or our temporal location. If we are suffering from amnesia, we need to figure out who we are. If we have an appointment to meet someone at a certain time, we might wish to know the current time. Sometimes, we 1

2 do not really care about our identity or temporal location except for the fact that reasoning about identity or temporal location might help us figure out which worlds are most likely to be actual. Anthropic reasoning is ubiquitous in both science and everyday life, but there are many difficult to analyze puzzles involving probabilistic anthropic reasoning such as Sleeping Beauty[8], Doomsday[13, 7], Lazy Adam[2], and Absent-Minded Driver[19]. We need a general theory that will allow us to analyze any anthropic reasoning problem. There already exists a well-known general rule for reasoning about self-locating belief. According to the Self-Sampling Assumption (SSA)[2], one should reason as if one were a observer randomly selected from some suitable reference class of observers. Here, by an observer is meant something that is capable of reasoning about self-locating belief. This is a very crude and imprecise definition of what it means to be an observer, but it will suffice for now. However we make the definition more precise, we want to be able to model normal adult humans as observers. The SSA is almost good enough but because observers can believe and desire different things at different times or make different choices at different times, we need the Strong Self-Sampling Assumption (SSSA)[2] according to which we should reason as if we were a randomly selected element of some suitable class of observer-moments. By an observer-moment we mean an ordered pair consisting of an observer and a time interval (and that time interval might just be a single point in time). We would use observer-moments to represent a time-slice of an observer. But there might be other relevant ways to split an observer into parts with each part being modelled as being capable of having its own coherent beliefs and desires and making its own coherent decisions. In that case, we would consider each of these parts to be a separate observermoment. For example, if someone is suffering from multiple personality disorder and thus might be modelled as consisting of several different personalities that exist at the same time, we might consider temporal slices of these personalities to be observer-moments. The SSSA is a perfectly fine general principle, but it is too underspecified. We need to know what is and what is not an observer. The random selection from the reference class presupposes the existence of a prior probability distribution on that reference class and the question arises what probability distribution should be used 1. We do not expect a theory of anthropic reasoning to tell us how to distribute prior probability among the many different possible worlds 2, but we do expect the theory to tell us how the prior probability that is given to a possible world is apportioned to the several observer-moments that inhabit that world. 1 And we, of course, also need to know how to choose reasonable reference classes. 2 However, it might be unrealistic to believe that we should separately analyze the nonanthropic and anthropic aspects of the problem of selecting a prior. 2

3 We need a precise theory that will tell us which prior probability distributions should be used by observermoments. Almost any reasonable theory of anthropic reasoning can be interpreted as an application of the SSSA or can be interpreted to be equivalent to the SSSA and the SSSA does allow us to apply standard techniques for reasoning about probabilities. We are not saying very much if we just say that we should use the SSSA. Even in the nonanthropic case, it is of limited use to just say that we are going to apply Bayesian methodology or that we will be doing Bayesian statistics if we do not say anything about how we choose our priors; we could (in the nonanthropic case) use reference priors[1], we could try to use a universal prior[10], we could use empirical Bayes[6], or we could do something else that would allow us to choose priors in a systematic way. In the anthropic case, we also need to be precise about how we generate our priors. The rest of this paper is organized as follows: In section 2 we introduce our centered possible worlds formalism and some notation that will make it easier to analyze wagers. In section 3 we justify a limited indifference principle[9]; if we know which world is actual, but we do not know which of two observer-moments we are and those observer-moments are in identical subjective psychological states, we should believe ourselves equally likely to be either observer-moment. This limited indifference principle is surprisingly difficult to justify. In section 4 we generalize our indifference principle to the case where the two-observer-moments are not in the same subjective psychological state but they still live in the same world. In this case, there is no question of our not knowing which of these two observer-moments we are. But we might still ask what we should believe if we do not take into account any anthropic information when estimating probability. This is a question about prior probability or about the random selection process used in the SSSA. We argue that if x and y live in the same possible world and have the same capacity for representing information and they both belong to the relevant reference class, then they should have equal probability of being chosen by the random selection process. Section 5 analyzes exactly what is and what is not an observer and then shows how we might derive a certain popular assumption, the Self-Indication Assumption[5] if we add enough ghost observer-moments to each possible world so that all worlds have the same number of observer-moments according to some reasonable way of counting observer-moments. Section 6 discusses the SIA in more detail and explains why we do not favor the SIA. In section 7, we discuss the issue of choosing reference classes. Once we see that if we use maximal reference classes, counterintuitive results can be derived both with and without using the SIA, we might notice that we can avoid some of our problems if we use minimal reference classes. But then, as section 7.1 points out, we will not be making full use of anthropic evidence that we actually do have. In 7.2 we 3

4 show that if minimal reference classes are used, we might be vulnerible to a collective Dutch Book. 7.2 also presents several arguments in favor of the use of almost maximal reference classes. One argument is a Dutch Book argument and another argument is a relative frequency argument. None of these arguments is incontrovertible and we have not really described how to handle the infinite case. We have to see which theory of anthropic reasoning works best in practice and which method for analyzing infinite scenarios works best. The testing of anthropic reasoning theories can be represented using standard Bayesian methodology(section 8). But we have to be careful about P (E T ), the probability that we would observe the evidence E that we actually observe given that the theory T of anthropic reasoning is correct. If the theory T is allowed to make use of knowledge of E in order to predict E, the fact that T can predict we observe E is not very surprising. Section 9 briefly describes why the mere fact that we seem to be atypical is not a reason to say that anthropic reasoning conflicts with observation. Section 10 relates our theory to the idea that updating is communication[16]. Section 11 discusses an application of anthropic reasoning to a puzzle (Lazy Adam) in which the decisions of one observer (Adam) determine whether other observers exist. Section 12 contains our conclusion and summary. 2 Formalism and Notation In this section, we first introduce our centered world formalism (2.1), then discuss when certain information might be considered irrelevant to estimating a posterior probability (2.2) and finally describe a notation for representing wagers (2.3). We might judge a theory of probabilistic reasoning by how successful rational agents are if they use the theory in order to determine how to solve decision problems; many decision problems can be represented as problems involving wagers. 2.1 The Centered Possible Worlds Formalism We let W represent the class of possible worlds of interest to us. W, in general, will not represent all worlds, only the interesting worlds, and if w W, w might not actually be a possible world, but an equivalence class of worlds. If the differences between two possible worlds are irrelevant 3, we consider them equivalent 3 To say that the difference between worlds v and w is irrelevant is to say the difference is neither directly relevant or indirectly relevant. The difference would be directly relevant if we cared about whether w is more likely to be actual than v. But even 4

5 and refuse to distinguish between them. All observer-moments of interest to us will be assumed able to reason coherently about which of the worlds in W is actual. But observer-moments do not have to be perfectly rational; some reasoning problems might be too hard for some of the observer-moments living in W. There might be some larger set V W of worlds such that some observer-moments living in W cannot reason coherently about which of the worlds in V is actual. Observer-moments of interest do have to be able to do more than just reason coherently about how likely they think various worlds in W are to be actual. They also have to be able to reason coherently while using only some of the knowledge that they actually have. Thus if an observer-moment actually knows K, and K is only part of her knowledge K and X W, she should be able to reason as if she only knew K and reason coherently about how likely it is that the actual world lies in the set X. We might not impose this requirement for all possible K, but we should require it for certain important K. One important knowledge set is the set of propositions known to be true by all observer-moments who live in worlds in W. But it is not just (bare) possible worlds that are of interest to us. We are really more concerned with centered worlds[14]. A centered world is just an ordered pair consisting of a (bare) world w and a center c. In general, many different kinds of center are possible, but in the centered worlds (w, c) that we analyze, c will be an observer-moment that inhabits world w. We let W represent the class of centered possible worlds of interest to us. We assume that every z W is obtained by enriching a world w W with a center c so that z = (w, c) and that for every w W, there exists at least one c such that (w, c) W. All z W are assumed to know 4 that they belong to W and to be able to reason probabilistically about who they are among the elements of W. They are assumed to know how to apply the SSSA. if the difference is not directly relevant, it might be indirectly relevant. If we are suffering from amnesia and want to know whether we are named George or Bill, we might not really be very interested in whether we live in a world v in which there are more people named George than Bill or a world w in which more are named Bill than George except for the fact that learning v rather than w is actual might tend to cause us to increase our estimate for how likely it is that we are named George. A precise definition of irrelevance will be provided in section When we refer to z W knowing or believing or getting utility from something, what we mean is the following: z is an ordered pair (w, c) where c is an observer-moment in world w. So we should be refering to the knowledge or beliefs or utility of c in w when we refer to something being known or believed by or the utility of z. A problem arises when we refer to what z believes or knows or to the utility of z; z might change her beliefs or desires. She might acquire knowledge or forget. Thus we cannot necessarily say about some proposition p that z believes that p or z does not believe that p. Both might be true but at different times. We also have the problem of in-between-believing[20]. At certain points in time, there might be no fact of the matter as to whether z believes that p. From some perspectives, she might be said to believe and from others, she might be said not to believe. We shall temporarily ignore these problems and assume that z has definite beliefs, desires, and knowledge. We also have the problem alluded to by Weatherson[21] of vague belief states. We can either just assume that observermoments have crisp and not vague beliefs or we can say that if it seems that it is vague whether some observer-moment z living in world w W believes p and does not believe q or vice versa, what is really happening is that w is really two different worlds in one of which z believes p and in the other of which z believes q. 5

6 We assume that all z W agree on a nonanthropic prior P. If w W, then P (w) represents the prior probability that world w is actual. It would be more precise to say that P (w) represents the conditional probability that w is actual given that the actual world lies in W (and not taking into account any knowledge that one observer-moment in W has and another does not have). Observer-moments might give nonzero prior probability to the possibility that some world outside of W is actual, but they might be uninterested in such a possibility or have difficulty reasoning about worlds outside of W. Thus strictly speaking observermoments in W do not necessarily know they belong to W, but they reason as if they knew they belonged to W. In order to analyze the set W, it is helpful to study W ; in order to analyze some subset V W, it will be helpful to study a certain set V of centered possible worlds. We shall generalize the star notation so that if V W, z V if and only if there exists v V such that there is a center c with z = (v, c) W. Thus V is the set of centered worlds in W that can be obtained by enriching a world in V with a center. In the case where V = {v} is a singleton set, we write v and not {v}. The star notation lets us go from bare worlds to centered worlds; in order to go in the reverse direction, if Z W, we define Ẑ W as the set of w W such that there exists c with (w, c) Z. If we think of (w, c) as part of w, then Ẑ might be thought of as the set of bare worlds which contain elements of Z. If Z = {z} is a singleton set, we write ẑ rather than Ẑ. In that case, ẑ is a singleton set consisting of just a single world; we shall be careless about distinguishing between the singleton set and the one world it contains. What a z W needs to do is the following: Given an interesting subset V W, compute the posterior probability that she belongs to V. First she needs a prior probability P z (V R z ) where R z W is the reference class used by z. The observer-moment is reasoning as if she were a random element of R z and the prior distribution P z used to do the random selection should satisfy a certain constraint that relates the nonanthropic prior P to the anthropic prior P z. We require for all A W that P z (A R z ) = P (A ˆR z ). We want the probability P (A ˆR z ) to be obtainable 5 by summing the probabilities of the observer-moments (in the reference class) that belong to worlds in A. Once z has a prior probability for V, she can compute her posterior P z (V R z K z ) where K z represents what z knows. The set K z is a set such that y K z if and only if z would say to herself that for all she knows she might be y. 5 when A contains only a finite number of observer-moments 6

7 2.2 Irrelevance We know that W might not actually represent possible centered worlds but instead represent equivalence classes of centered worlds and that if the difference between two worlds is irrelevant we might consider them equivalent. We should be more precise about what irrelevance means. So let us start with an actual set W of (bare) possible worlds and an actual set W of centered worlds rather than equivalence classes of worlds and centered worlds. We consider some observer-moment z W who wants to estimate P z (A K z ) for some A W and ask ourselves when it is acceptable for z to consider certain (bare) worlds and consider certain centered worlds equivalent. Let represent an equivalence relation on W. Let represent an equivalence relation on W. We would normally expect there to exist some relationship between these two equivalence relations. If an observer-moment i can exist in two different possible worlds, v and w, we might consider (v, i) and (w, i) to be counterparts of each other and then if v w, we would expect (v, i) (w, i); more generally we might define a natural counterpart relationship between observer-moments living in different (bare) worlds and if i living in v is a counterpart of j living in w and v w, we would expect (i, v) (j, w). If V W, then we write V to represent the closure of V under. Thus if w is a (bare) world, w V if and only if there exists a v V with v w. If Z W, we write Z to represent the closure of Z under. Thus z Z if and only if there exists y Z with y z. If C is any set and R is an equivence relation on C, then we can generate the set G of equivalence classes of C modulo R. We have g G if and only if there exists a c C such that g is the set of elements d C with d R c. If B C, then B mod R refers to the set of g G with g B. We shall make statements about closures Z and V, but it should be easy to translate these statements into statements about equivalence classes Z mod and V mod. For example, in order to evaluate a probability for Z mod, it suffices to evaluate the probability of Z. It just simplifies the exposition to use closures rather than equivalence classes. In order to work with closures, we need to insure that P z is defined on all of W not just on R z, but that is simple enough if we just specify that for all X W with X R z =, P z (X) = 0. We might assume that the set A whose posterior probability z wants to estimate is closed under. Then what z really cares about is whether P z (A K z ) = P z (A (K z ) ) 7

8 and thus we care whether P z (A K z ) P z (K z ) = P z(a (K z ) ). P z ((K z ) ) Assuming that P z (A K z ) 0, that means that we care about whether P z (K z ) P z ((K z ) ) = P z(a K z ) P z (A (K z ) ). In words, this is requiring that the probability that a random element of the closure (K z ) of the knowledge set is actually an element of the knowledge set K z be independent of whether that random element is also an element of A. If this condition is true, than z can refuse to distinguish between equivalent bare worlds and equivalent centered worlds when estimating the probability of A. Of course in the previous paragraph, z is using the fact that the closure of her knowledge state is (K z ) and there might exist an observer-moment y z such that (K y ) (K z ) ; in that case the difference between equivalent observer-moments is not totally irrelevant even if it is basically irrelevant for z who only cares about the posterior probability of A. If the only subsets of W about which z cares are sets B that are closed under and the differences between equivalent uncentered worlds and equivalent centered worlds is basically irrelevant with respect to any such set B and for all z y, (K z ) = (K y ), then from z s point of view, the differences between equivalent worlds and equivalent centered worlds is totally irrelevant. 2.3 Wagers We also need some special notation to represent wagers. A wager is represented by a function f from W to R where R represents the real numbers. If z W, f(z) is the amount of utility that z gains by accepting rather than rejecting the wager. We assume that the amount gained by z is independent of how other y W decide when offered the wager f and also independent of any other wagers that might be offered. We let f have domain all of W so that no useful information about one s location within W can be gained from the fact that one is offered a certain wager. The return from a wager is a utility value rather than a monetary value because utility is not necessarily linear in monetary return. We assume all z W try to maximize their expected utility y K z P z (y)f(y) 6. We shall have occasion to add the utilities of two y and z such that K y K z. If K y = K z, then z might have to compute an expected utility and thus might have to be able to compare y utility and z utility. She 6 We are assuming that there is no overlap: If x y are two observer-moments in W, then it is not possible to be both x and y. Thus if x and y represent time-slices of the same observer in the same possible world, the time-intervals during which x and y live are disjoint. 8

9 will have to be able to measure y utility and z utility on a common scale so that one unit of y utility is as valuable as one unit of z utility. But if K y K z, no one actually needs to measure y utility and z utility on a common scale in order to decide what to do. But we can ask how an observer-moment would make decisions if she did not know whether she was y or z but knew that she had a certain probability of being y and a certain probability of being z. This really means that she has a certainly probability of being someone just like y and a certain probability of being someone just like z 7 because anyone who actually might be y knows she is not z and vice versa. But in any case it should be possible to perform an act of imagination and imagine that one does not know whether one is y or z and then try to figure out how one would make decisions when not sure whether one is y or z. Even if it really does not make sense to compare y utility and z utility if K y K z, it is not impossible that there might be some preferred way to scale the utilities of y and z so that one unit of y utility and one unit of z utility are equally valuable. Theories of anthropic reasoning should work if it turns out that it makes sense to add the utilities of observer-moments y and z with K y K z and it will make sense to add the utilities if it is possible to measure the utilities of y and z using a common scale. 3 A Limited Indifference Principle In this section, we argue for a limited indifference principle that if x, y W with ˆx = ŷ and K x = K y, then P x (x) = P x (y)[9] 8. This is a very limited principle. It only says that if there are two observer-moments who live in the same possible world and who are in the same subjective psychogical state and for all we know we might be one of these observer-moments, then we are no more likely to be one of them than to be the other one. We shall present several possible arguments for this principle. Some of these arguments are more convincing than others but we do need a principle that will enable us to compute Px(x) P x(y). When trying to construct an argument for our limited indifference principle, we must keep in mind that the argument should not be too easily generalizable. We do not expect a human observer-moment that lasts ten million seconds to have the same prior probability as one that lasts one second; these two observer-moments will not be in the same (relevant) subjective psychological state (or sequence of states). We also need to be very clear about how limited our principle is. It is certainly not true that if K x = K y and x and y inhabit different worlds that they necessarily have the same prior probability. Nor do we have 7 Someone who has preferences and experiences that are very similar to the preferences and experiences of z. 8 Elga s[9] discussion refers to observers, rather than observer-moments, but his principle is in essence the same as ours. 9

10 the principle that if x, y, z live in the same world and are in the same subjective psychological state, then P z ({x, y}) = 2P z (z). The problem is that there might be overlap. In the worst case x and y might be the same observer-moment. Or they might share some of their computational hardware. Or x and y might not be making independent observations of their environment so there might be a sense in which if I am x, then I am also in part y. This happens with human observer-moments if x and y are observer-moments that belong to the same observer 9 and y occurs immediately after x and thus regardless of how radically and how quickly the external environment might be changing, it takes time to correct the obsolete information y has received from x. The first argument for the limited indifference principle is that it is a simple principle that is not obviously absurd and it is hard to think of another principle that is equally simple and that is workable. If K x = K y and ˆx = ŷ and Px(x) P x(y) 1, what is the ratio to be? We might say that P x(y) and P x (x) should depend on the lengths of the briefest descriptions of y and x in some canonical language. But then we have the added complexity of discovering the ideal canonical language. The problem with this argument is that it is too easily generalizable to the case where x and y live in different possible worlds. We certainly do not want to say that if K x = K y with ˆx = v w = ŷ and w is a world in which with a few exceptions every observer is a brain in a vat but v is a world more like the actual world, that P x (x) should equal P x (y). We are not all that likely to be a brain in a vat. The second argument is a wagering argument. We can without loss of generality restrict to the case where both x and y know they both belong to a certain world w and they both know they are either x or y but they do not know which one and we assume there is no overlap. If there were overlap, we could just find some reasonable way of modifying our P x so that P x (x x or y) + P x (y x or y) = 1 Because K x = K y, we must have P x = P y. Assume that P x (y) = rp x (x) with r > 1. So P y (y) = rp y (x). If there is a wager f such that f(y) = 1 and f(x) = s with 1 < s < r while f(z) = 0 unless z is either x or y, then both x and y will compute an expected value of r r+1 s r+1 > 0 and accept the wager. But then it is inevitable that the total return is 1 s < 0. So it seems it would have been better if both x and y had rejected the offer. And since x and y are in the same subjective psychological state, they would either both accept or both reject. Thus the best option is for them to both reject and that appears to show that if P x (y) = rp x (x), then r should not be greater than 1. A very similar argument would show that r should not be less than 1. Thus P x (y) = P x (x). This wagering argument is too similar to an argument that in the Prisoners dilemma scenario it is rational 9 We say that x, y W belong to the same observer if they are temporal slices of the same observer in the same possible world. 10

11 for both prisoners to cooperate because it is better that they both cooperate than that they both defect. However, each individual prisoner is better off if she defects regardless of what the other prisoner does. The mere fact that one prisoner cooperates does not force the other to cooperate. In our scenario both x and y can say: I am much more likely to be y than x. So it makes sense for me to accept a wager f with f(y) = 1 and f(x) = 2. I know the other guy (x if I am y and y if I am x) thinks the same thing but she is mistaken. It is not desirable that the other guy accept, but that should not stop me from accepting because I am more likely to be y. It is certainly consistent to say that x and y should have equal prior probability but it is also consistent to say that y should have ten times as much prior probability. Yet it seems intuitively reasonable if we have to choose between two possible solutions to the problem of choosing priors that we prefer the one that results in greater total utility and greater average utility. This argument about what is intuitively reasonable does not generalize to the case where x and y live in different worlds; we really only care about the total or average utility of the observer-moments who actually exist. If x and y live in the same world we can assume that we have scaled the utility functions of x and y in such a way that it makes sense to compute a simple sum or average of the utilities of x and y. If ˆx ŷ, then we would be more interested in a weighted sum or average of the utilities of x and y where the weights of x and y are proportional to P (ˆx) and P (ŷ) respectively. Our third argument is that there is a sense in which it is not consistent to say that y should have greater (for example, three times as much) prior probability, than x. If I am x (or y) and I say that The other guy has only a probability of 1 4 of being y even though the other guy is in the same subjective psychological state as I am and I believe I have a probability of 3 4 of being y, there is no obvious justification for the difference between what I believe is my probability of being y and what I believe is the other guy s probability. The third argument does not generalize to the case where only one of the observer-moments x, y exists. In that case, there would be one big difference between me and the other guy: I exist and she doesn t. The fourth argument is a simplified version of an argument of Elga s[9]. The argument involves comparing three different scenarios and is based on the assumption that similar scenarios should be analyzed similarly. The argument is not quite convincing but it is still worth analyzing. In the first scenario, there is only one possible world and in that world, there are only two observers, Al and AlDup (a duplicate of Al). Originally, there was just one observer, but then at a certain point t 0 in time, a duplicate of Al appeared. At any point in time subsequent to t 0, Al and AlDup are in the exact same subjective psychological state. At some time t 1 after t 0, Al wants to estimate the probability that he is Al rather than AlDup. (Of course, AlDup also wants to estimate the probability that he is Al.) We 11

12 might analyze the scenario as containing two relevant observer-moments, Al at time t 1 10 and AlDup at time t We would like to show that Al at time t 1 should believe the probability that he is Al to be.5. If we could demonstrate this, it should not be too hard to show that our limited indifference principle that P x (x) = P x (y) if K x = K y and ˆx = ŷ should be true in general. In order to help us analyze this first scenario, we consider a second scenario in which there are two possible worlds, H and T. These worlds differ only in how an unfair coin tossed by a robot lands. If the coin lands heads, H is actual (probability.1) and if the coin lands tails, T is actual (probability.9). In both worlds Al and AlDup are the only observers and in both worlds at any time after time t 0, Al and AlDup are in the exact same subjective psychological state. The duplication and coin-tossing are assumed to be completely independent processes. The coin-tossing does not affect the subjective psychological state of either Al or AlDup. We might analyze this scenario as containing four relevant observer-moment Hal (Al at time t 1 in the heads world), Tal (Al at time t 1 in the tails world), HalDup (AlDup at time t 1 in the heads world) and TalDup (AlDup at time t 1 in the tails world). Since the coin-tossing is entirely independent of the duplication, in order to show that in the first scenario Al should believe the probability he is Al equals.5, it suffices to show that in the second scenario Hal (as well as Tal, HalDup, and TalDup) should believe that the probability that he is Al (i.e. that he is either Hal or Tal) to be.5. We consider a third scenario, that is exactly like the second scenario except that at a certain time t 2 later than t 1 one of the two observers goes into a coma. In H, it is AlDup who goes into a coma and in T, it is Al. At some time t 3 later than t 2, Al in the world H wants to estimate the probability that he is Al. Al at time t 3 in H is in the same subjective psychological state as AlDup at time t 3 in T. So AlDup at t 3 in T also wants to estimate the probability that he is Al. We might analyze the third scenario as containing six relevant observer-moments. We have the same four relevant observer-moments that we had in the second scenario: Hal, Tal, HalDup, and TalDup. These are all in the same subjective psychological state. We also have Hal2 (Al at time t 3 in H) and TalDup2 (AlDup at time t 3 in T). Hal2 and TalDup2 are in the same subjective psychological state but their subjective psychological state is different than the state of Hal, Tal, HalDup, and TalDup. If we assume that Hal2 uses {Hal2, TalDup2} as his reference class, clearly Hal2 should believe that the probability that he is Hal2 is.1. The other reasonable choice of reference class is the class consisting 10 t 1 might be a short time-interval rather than just a point in time. 11 The reason we might consider these two observer-moments to be the only relevant observer-moments is that nothing essential changes if we assume that both Al and AlDup are unconscious except at t 1. 12

13 of all six observer-moments. Because the coin tossing is assumed irrelevant, we would like to assert that P Hal2 (Hal2 H) = P Hal2 (TalDup2 T ) because that would allow us to conclude that even if the larger reference class is used, Hal2 s posterior probability estimate for his being Hal2 should still be.1. However, unfortunately, it is perfectly consistent to give Hal2 and Hal twice the prior probability of HalDup and give Tal twice the prior probability of TalDup and TalDup2. (Regardless of how the coin falls originals have twice the prior probability of duplicates.) Let us just assume that Hal2 should conclude that the posterior probability of H is.1. If we could assume that P Hal (Hal Hal or TalDup) = P Hal2 (Hal2 Hal2 or TalDup2) =.1, simple algebra would show that P Hal (Hal Hal or HalDup) =.5. And then given the assumption that similar scenarios should have similar analyses, we would have the result that in the first scenario, Al should believe that he is as likely to be Al as AlDup. Unfortunately, Hal and Hal2 are different and it need not be the case that P Hal (Hal Hal or TalDup) = P Hal2 (Hal2 Hal2 or TalDup2). It would be the case if a certain continuity assumption is true, but it is not clear that the continuity assumption in question is any more obvious and any less in need of proof than our limited indifference principle. Assuming that our four arguments demonstrate the truth of the indifference principle that if K x = K y with ˆx = ŷ, then P x (x) = P y (x), we find it natural to believe that if K x = K y with ˆx = ŷ, then P z (x) = P z (y), for any z W such that x, y R z. A general argument for this result is that P z(x) P z(x)+p z(y) is z s estimate of how likely she should think it is that she is x given that she knows that she is either x or y 12. But someone who knows that she is either x or y would be in knowledge state K x and thus would believe it as likely that she be x as that she be y. Therefore, we should have P z(x) P z(x)+p z(y) =.5 and P z(x) = P z (y). We would also like to say something about the ratio of the prior probabilities of x and y when x and y live in the same possible world but are not in the same subjective psychological state. 4 Assuming ˆx = ŷ what should be the value of P z(y) P z (x) when y, x R z? If x and y are in different knowledge states (K x K y ), then there is no question of an observer-moment not knowing whether she is x or she is y, but the ratio Pz(y) P z(x) simple scenario. still matters. To see why, consider the following The Fundamental Scenario: There are only two possible worlds, v and w. If they ignore anthropic information, the observer- 12 We are still assuming that overlap is not a problem. 13

14 moments in v and w would have no reason to think one of these worlds more likely to be actual than the other world. But observer-moments do have anthropic information available. In both worlds, there are exactly two possible subjective psychological states, A and B. We can represent v as containing c observer-moments in state A and d observer-moments in state B. The world w consists of e observer-moments in state A and f in state B. The numbers c, d, e, f are all finite. We have no problem with overlap. All the different observer-moments in each world are genuinely distinct observer-moments and thus for example, the c observer-moments in world v who are in state A are all distinct and do not share resources or inhibit each other s capacity for believing and desiring and accepting or rejecting offers to wager. Every observer-moment needs to estimate the probability the actual world is v taking into account the anthropic information she actually does have. We call this scenario fundamental because if we know how to analyze this scenario, we know how to analyze most scenarios that can be represented as having only a finite number of possible worlds with observers and in which every world has no more than a finite number of observer-moments. In this scenario, all observer-moments are assumed to know all the details of the scenario and to know which subjective psychological state they are in, but in general, they do not know which world they inhabit and have only limited information about their identity and temporal location: They just know that they are in state A or they know they are in state B. Assume all observer-moments in the same possible world have equal prior probability and let z be an observer-moment who is in state A. We assume that z uses a reference class consisting of all c + d + e + f observer-moments. Not taking into account her anthropic information, z would say that the prior probability that both v is actual and she is in state A is P z (A and v) = ( c c+d )( 1 2 ). The prior probability that she is in state A and that w is actual is ( e a posterior probability of e+f )( 1 2 c c+d c c+d + e e+f ). After taking into account her knowledge that she is in state A, we obtain for v being actual. This simplifies to c(e+f) c(e+f)+e(c+d). If, however, z had assumed that observer-moments in state A had many times more prior probability than observer-moments in state B, she would arrive at a very different posterior probability. If she assumed that observer-moments in state A had infinitely many times as much prior probability as those in state B, she would arrive at a posterior probability of.5. A natural hypothesis about Pz(x) P z(y) in the case when ˆx = ŷ and x, y R z is that prior probability should be proportional to cognitive complexity. This is implicitly assumed in, for example, [2]. Since we want P z (x) = P z (y) if K x = K y and ˆx = ŷ, we shall assume that observer-moments in the same subjective 14

15 psychological state have the same amount of cognitive complexity. We do not have to make the assumption, but if we did not make the assumption, instead of discussing the cognitive complexity I(x) of a certain observer-moment x, we would have to discuss the average complexity of an observer-moment in world ˆx and subjective psychological state K x. So instead of saying that P z (x) = P (ˆx) I(x) I(ˆx) where I(x) represents the complexity of x and I(ˆx) represents the cognitive complexity of the whole set of observer-moments living in the same world as x and we assume that the total complexity is finite, we could use the same formula but I(x) would have to represent the average complexity of the observer-moments in ˆx K x. Our arguments and our exposition can be simplified if we assume that if K x = K y, then I(x) = I(y). One measure of the cognitive complexity of an observer-moment z is the amount of (relevant) information I(z) that she is capable of representing. This suggests that we should have Pz(x) P = I(x) z(y) I(y) I(x) and I(y) are both finite. in the case where The basic reason is that we believe our information theoretic rule to be correct is that it is simple and intuitively appealing. We also have a wagering argument for our rule and an argument based on the concept of indecomposable or atomic moment and the idea that we should be able to represent W as a union of independent atomic moments. We first give our wagering argument. We assume that W and W are really equivalence classes of worlds and centered worlds and that if we just specify z W, we have not specified certain potentially important information about z; we have not specified which decision problems 13 z needs to solve. Maybe she does not need to explicitly represent which problems she is trying to solve but she does need to devote cognitive resources to solving these problems. Our key assumption is that the total amount of cognitive resources that an observer-moment can devote to solving decision problems is proportional to the amount of information she is capable of representing about who she is among the observer-moments in W. Or we might just assume that the amount of information that z can store about which problems she is trying to solve is proportional to the amount of information she is capable of representing about who she is among the elements of W 14. In any case, we are assuming that computational resources that can be used to solve decision problems are scarce. In order to solve a certain decision problem, an observer-moment might need to make use of her posterior probability estimate for how likely it is that she belongs to some set Z W. Even if P z (Z K z ) is trivial to compute, some computational resources will be spent when z computes the probability and then uses the probability to optimize her decision-making. 13 Wagering problems are decision problem. We might also regard probability estimation problems as decision problems in which an observer-moment is trying to maximize some kind of epistemic utility. 14 Thus we are treating information storage space as the only scare resource. 15

16 Given that scarcity of computational resources exists, we will explain why the scarcity matters. We will use a wagering argument. Not every decision problem is a wagering problem but our argument can be generalized to apply to decision problems that are not wagering problems. We are interested in P z, a prior that supposedly does not take into account any information that z has and other observer-moments in W do not have. Thus we shall not take into account which wagering problem z is trying to solve. We just assume it is some random wagering problem. We assume because computational resources are scarce and the fact that z is a rational agent is only demonstrated when z is trying to solve some decision problem and because there are so many possible decision problems that she might need to solve, that the probability that z will have the computational resources available to solve a given randomly chosen decision problem is small 15 and we assume the probability is proportional to the amount of information she is capable of representing about her location among the observer-moments in W. This assumption is most reasonable for simple, easy to describe decision problems but more complex problems can be represented as a sequence of simpler problems. We want the observer-moments in W to choose their prior probabilities in such a way as to optimize expected utility when faced with a random wagering problem D. When computing this expected utility, we will ignore those observer-moments who do not have the resources available to solve the wagering problem. If we could restrict to the case where in each w W, there is at most one y w who has the resources to solve D, then the probability that a given z will actually be trying to solve D is the product of P (ẑ), the probability that z lives in the actual world, and I(z) I(ẑ). And that would be a reason for prior probabilities to be proportional to amount of information represented. In reality, there might be some worlds where many observer-moments have the resources to solve D, but we might pretend that D comes in several different variants. There is no essential difference between the variants, between the different ways that a decision problem might be formulated. We might describe our situation as one in which all an observer-moment knows is that she is dealing with a random variant of a random decision problem. If there are enough variants, then it is quite likely to be true that in each world only at most one observer-moment will have the resources available to solve a specific random variant of D. And we have to analyze each variant differently since each variant is a different problem and for each variant there is a different set of observer-moments who have the available cognitive resources to deal with the variant. Introducing these imaginary variants should not affect which prior P z should be used. But with the help of these variants we could see why prior probability should be proportional to cognitive complexity. 15 We are assuming that which other decision problems need to be solved by z is determined by some stochastic process. 16

17 This argument might seem to similar to an argument that in the fundamental scenario if c = e = 1 (in both v and w, there is one observer-moment in state A) and f = , then an observer-moment z in state A should believe the two possible worlds equally probable. But this argument for P z (v) = P z (w) would be ignoring the fact that observer-moments in state B exist in world w. In our wagering argument, we treat observer-moments who do not have the resources to handle a random problem as if they were not observer-moments. We are treating them as if they do not exist 16. We are only using our wagering argument to determine a prior. We need some way of constraining our prior. We are following the general philosophy of making our prior as uninformed as possible, taking into account as little as possible. If we actually have more information, we can conditionalize. Thus we will have wagering arguments constraining what P z is, but we will not try to provide a wagering argument to constrain the posterior P z ( K z ). We also have another justification of our information-theoretic rule based on indecomposability (i.e. atomicity). To understand why we care about indecomposability, reflect about the fact that some observer-moments live too long to be analyzed as unified observer-moments and should really be decomposed into sets of shorter-lived observer-moments. There is a sense in which it is difficult to conceive of a human observermoment z that lasts ten thousand seconds as having definite beliefs or making definite decisions or being in a definite subjective psychological state. In ten thousand seconds, the external environment can change drastically. There can be justifiable drastic changes of relevant belief during those ten thousand seconds. It might be quite misleading to reason about z as if she had a definite knowledge state K z. Wagering or decision-theoretic arguments based on the assumption that observer-moments have definite beliefs and make definite decisions based on those beliefs and a computation of the action that leads to greatest expected utility might not really be applicable (even approximately) to an observer-moment like z. A wager offered to z during the last three hundred seconds of her life might be something about which she was ignorant for most of her life. There is a sense in which we should not try to model z as if she were a single rational agent with definite desires and beliefs. It might be true just by happenstance that the long-lived observer-moment z does have definite desires and beliefs, but if z lives long enough, it could easily be the case that one part of z is exposed to different evidence than another part of z and thus different parts of z have different beliefs. If it makes sense to split 16 This is very different from saying we want to compute the posterior probability of some specific set A and then ignoring observer-moments who do not need to know this probability. When we know that we need to know the probability of A and other observer-moments do not need to know this probability, we know some information that is not common knowledge to all z W. But we can picture all observer-moments as knowing that they are trying to solve some random decision problem. 17